Regenerative processes in the infinite mean cycle case

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the wai...

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Bibliographic Details
Published in:Journal of applied probability 2001-03, Vol.38 (1), p.165-179
Main Authors: Mitov, K. V., Yanev, N. M.
Format: Article
Language:English
Subjects:
60F05
60K05
alternating regenerative processes
alternating renewal processes
Distribution functions
Exact sciences and technology
infinite mean cycles
Infinity
Limit theorems
limiting distributions
Mathematical theorems
Mathematics
Perceptron convergence procedure
Probability
Probability and statistics
Probability theory and stochastic processes
Random variables
Research Papers
Sciences and techniques of general use
Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)
Studies
Theory
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Summary:A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/996986651